solar plasmas T.V. Zaqarashvili and M.L. Khodachenko Space Research - PowerPoint PPT Presentation

Two-fluid MHD approach for partially ionized solar plasmas T.V. Zaqarashvili and M.L. Khodachenko Space Research Institute of Austrian Academy of Sciences, Graz, Austria

Solar atmospheric model FAL93-3 model (Fontenla et al. 1993)

Solar atmospheric model n e – electron number density n H – neutral hydrogen number density n He – neutral helium number density Plasma is only weakly ionized in the photosphere, but becomes almost fully ionized in the transition region and corona. FAL93-3 model (Fontenla et al. 1993)

Effects of neutral atoms in the solar atmosphere Neutral atoms may change the plasma dynamics through collisions with charged particles: Damping of MHD waves (Braginkii 1965, De Pontieu et al. 2001, Khodachenko et al. 2004, Leak et al. 2006, Forteza et al. 2007, Soler et al. 2009, 2010, Carbonell et al. 2010, Singh and Krishan 2010, Zaqarashvili et al. 2011a,b) ; Formation of spicules (Haerendel 1992, De Pontieu & Haerendel 1998, James & Erdélyi 2002, James et al. 2004); Influence on energy flux of Alfvén waves in the photosphere (Vranjes et al. 2008); Generation of electric currents and plasma heating (Sen and White 1972, Khodachenko & Zaitsev 2002, Fontenla et al. 2008, Gogoberidze et al. 2009, Krasnoselskikh et al. 2010, Khomenko and Collados, 2012a,b); Emerging magnetic flux tubes (Leake & Arber 2005, Arber et al. 2007); Influence on resonant absorption (Soler et al. 2009). Influence on Kelvin-Helmholtz instability (Soler et al. 2012).

Three ree-Fluid Fluid equations ations   n     e ( n V ) 0 ,  e e We consider partially ionized plasma, t which consists of electrons (e), protons   n     i ( n V ) 0 , (i) and neutral hydrogen (n).  i i t   n     n ( n V ) 0 ,  n n t             V 1                  e m n ( V ) V p en E V B R ,    e e e e e e e  e  e   t c             V 1                  i m n ( V ) V p en E V B R ,    i i i i i i i  i  i   t c (Braginski 1965)        V             n m n ( V ) V p R ,    n n n n n n n   t        3 T                e ( ) : , n k V T p V V q Q  e e e e e e e e e   2 t        3 T                i n k ( V ) T p V : V q Q ,  i  i i  i i i i i i 2 t        3 T                n n k ( V ) T p V : V q Q ,  n n n n n n n n n   2 t    p n kT , p n kT , p n kT , e e e i i i n n n   q Q R is the change of impulse, is the heat flux density, is the heat production. a a a

Maxwell xwell eq eq uations tions The description of the system is completed by Maxwell equations    1 B     E ,  c t    4     B j , c where         j en V V e e i is the current density and   B   0 . Plasma is supposed to be quasi neutral n  n . e i

Imp mpulse lse change nge and heat producti duction on Impulse change and heat production can be expressed as (Braginskii 1965)                 , R V V V V e ei e i en e n                 R V V V V , i ie i e in i n                 R V V V V , i ne n e ni n i                 Q V V V V V V , e ei e i e en e n e                 Q V V V V V V , i ie i e i in i n i     ,             Q V V V V V V n ne n e n ni n i n    are coefficients of friction between different sort of particles. ab ba For time scales longer than ion-electron collision time, the ion-electron gas can be considered as a single fluid. Then, any additional sort of neutral atoms can be treated as a separate fluid.

Friction ction coefficients icients The coefficient of friction between ions and electrons can be expressed as (Braginskii 1965)   4 4 2 e n n m   i e ie ,   ie 3 / 2 3 m kT ie e  where is the Coulomb logarithm . The coefficient of friction between ions and neutral hydrogen atoms is (Braginskii 1965) 8 kT    n n m ,  in i n in in m in  m where is ion-hydrogen collision cross-section and is the reduced mass. in in For elastic collision, the ion-hydrogen collision cross-section is (Braginskii 1965)   ,    r  2 2 r in i n which approximately equals atomic cross section.

Colli llision ion frequ quencies ncies Ion-electron collision frequency is expressed by    4 4 2 e n    ie i .   ie 3 / 2 m n 3 m kT  e e e e   in , Ion-neutral collision frequency is often expressed as in m n i i    in . while neutral-ion collision frequency is often expressed as ni m n n n According to this formulation, ion-neutral collision frequency is different than neutral-ion collision frequency. But, the mean collision frequency between ions and neutral atoms should be a single value due to physical basis. From simple equations of motion of ions and neutrals one can derive the equation for relative velocity between ions and neutrals             V V 1 1        i n V V .    in i n   t m n m n i i n n This equation gives a single value for the ion-neutral collision frequency (Zaqarashvili et al. 2011)   1 1 kT            2 ( n n ) .    in ni in i n in   m n m n m i i n n i

Collis ision ion frequen enci cies es in the solar atmosphe sphere re Let us use FAL93-3 model (Fontenla et al. 1993) for temperature and number densities: z=0: z=900: z=1900: T=6520 K T=6000 K T=8900 K 13 cm -3 11 cm -3 11 cm -3 n e =7.67 10 n e =2.60 10 n e =1.27 10 13 cm -3 11 cm -3 11 cm -3 n i =5.99 10 n i =2.43 10 n i =1.20 10 17 cm -3 13 cm -3 11 cm -3 n n =1.18 10 n n =8.95 10 n n =1.79 10  8 Hz 10 6 Hz 6 Hz =8 10 3.7 10 ie  6 Hz 6.2 10 3 Hz Hz =8.6 10 24 in Ion-electron collision frequency seems to be always much higher than ion-neutral collision frequency in the whole atmosphere. However, this statement may be overestimated in the lower photosphere, where significant number of heavy ions exists.

Two-Fluid luid MHD HD equations uations   n     i ( n V ) 0 ,  i i t   n     n ( n V ) 0 ,  n n t                V 1                 i en m n ( V ) V p j B j V V ,    i i i i ie in en i n   t c en e              V               n en m n ( V ) V p j V V ,    n n n n n in en i n   (Zaqarashvili et al. 2011) t en e          p                      2 ie ei ( V ) p p V 1 j 1 V V V  i ie ie i in i n i 2 2 t e n e                j p j                      e 1 V V V p 1 q q , en e n e e i e en en e e               p                                n ( V ) p p V 1 V V V 1 V V V 1 q ,  n n n n in i n n en e n n n t   p p p is the pressure of electron-ion gas. ie i e